Triangles in a Cube - Solution
by
Erik Oosterwal
The first step in solving this problem is to figure out how many different
triangles can be made using the vertices of a cube. Since there are
8 vertices, the first point can be picked from one of 8 vertices, the second
point can be picked from the 7 remaining vertices, and the last point of
the triangle can be picked from the 6 remaining vertices. There are, therefore,
8•7•6 = 336 ways to draw a triangle on (or inside) a cube, when
you include rotations. Eliminating rotations and mirror images, we
are left with 56 unique triangles.
Of those 56 triangles, only 8 are regular triangles. To draw a regular
triangle in a cube, the first two points must fall on opposing diagonals
of a single face. The last point is taken from one of the two points
on the opposite face on a corner that does not share an edge with the first
two points. The picture below shows 4 of the 8 possible regular
triangles:
The remaining triangles are all right triangles. Therefore the probability
of getting a regular triangle is 8/56 = 0.14286..., and the probability of
getting a right triangle is 48/56 = 0.85714...
Copyright E. Oosterwal - 2004
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